On stability of the P n mod / P n element for incompressible flow problems

Petr Knobloch

Applications of Mathematics (2006)

  • Volume: 51, Issue: 5, page 473-493
  • ISSN: 0862-7940

Abstract

top
It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of n th order accuracy in the energy norm called P n elements. For n ≤ 3 we show that the stability condition holds if the velocity space is constructed using the P n elements and the pressure space consists of continuous piecewise polynomial functions of degree n .

How to cite

top

Knobloch, Petr. "On stability of the $P^{\rm mod}_ n/P_ n$ element for incompressible flow problems." Applications of Mathematics 51.5 (2006): 473-493. <http://eudml.org/doc/33263>.

@article{Knobloch2006,
abstract = {It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of $n$th order accuracy in the energy norm called $P_n^\{\}$ elements. For $n\le 3$ we show that the stability condition holds if the velocity space is constructed using the $P_n^\{\}$ elements and the pressure space consists of continuous piecewise polynomial functions of degree $n$.},
author = {Knobloch, Petr},
journal = {Applications of Mathematics},
keywords = {nonconforming finite element method; inf-sup condition; incompressible flow problem; nonconforming finite element method; inf-sup condition; incompressible flow problem; incompressible flow},
language = {eng},
number = {5},
pages = {473-493},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On stability of the $P^\{\rm mod\}_ n/P_ n$ element for incompressible flow problems},
url = {http://eudml.org/doc/33263},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Knobloch, Petr
TI - On stability of the $P^{\rm mod}_ n/P_ n$ element for incompressible flow problems
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 5
SP - 473
EP - 493
AB - It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of $n$th order accuracy in the energy norm called $P_n^{}$ elements. For $n\le 3$ we show that the stability condition holds if the velocity space is constructed using the $P_n^{}$ elements and the pressure space consists of continuous piecewise polynomial functions of degree $n$.
LA - eng
KW - nonconforming finite element method; inf-sup condition; incompressible flow problem; nonconforming finite element method; inf-sup condition; incompressible flow problem; incompressible flow
UR - http://eudml.org/doc/33263
ER -

References

top
  1. 10.1093/imanum/22.2.307, IMA J. Numer. Anal. 22 (2002), 307–327. (2002) MR1897411DOI10.1093/imanum/22.2.307
  2. 10.1051/m2an:2000111, M2AN, Math. Model. Numer. Anal. 34 (2000), 953–980. (2000) MR1837763DOI10.1051/m2an:2000111
  3. 10.1137/0720048, SIAM J. Numer. Anal. 20 (1983), 722–731. (1983) MR0708453DOI10.1137/0720048
  4. Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. (1991) MR1115205
  5. Basic error estimates for elliptic problems, In: Handbook of Numerical Analysis, Vol. II: Finite Element Methods (Part  1), P. G.  Ciarlet, J.-L. Lions (eds.), North-Holland, Amsterdam, 1991, pp. 17–351. (1991) Zbl0875.65086MR1115237
  6. Conforming and nonconforming finite element methods for solving the stationary Stokes equations  I, Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973), 33–76. (1973) MR0343661
  7. 10.1090/S0025-5718-1989-0958870-8, Math. Comput. 52 (1989), 437–456. (1989) MR0958870DOI10.1090/S0025-5718-1989-0958870-8
  8. Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. (1986) MR0851383
  9. Parallele Lösung der inkompressiblen Navier-Stokes Gleichungen auf adaptiv verfeinerten Gittern, PhD. Thesis, Otto-von-Guericke-Universität, Magdeburg, 1997. (1997) 
  10. 10.1007/s00607-002-1444-2, Computing 68 (2002), 313–341. (2002) MR1921254DOI10.1007/s00607-002-1444-2
  11. 10.1007/s00791-004-0127-2, Comput. Visual. Sci. 6 (2004), 185–195. (2004) MR2071439DOI10.1007/s00791-004-0127-2
  12. New nonconforming finite elements for solving the incompressible Navier-Stokes equations, In: Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH  2001, F.  Brezzi et al. (eds.), Springer-Verlag Italia, Milano, 2003, pp. 123–132. (2003) Zbl1283.76033MR2360713
  13. On the inf-sup condition for the  P 3 m o d / P 2 d i s c element, Computing 76 (2006), 41–54. (2006) MR2174350
  14. 10.1137/S0036142902402158, SIAM J.  Numer. Anal. 41 (2003), 436–456. (2003) MR2004183DOI10.1137/S0036142902402158
  15. Parallele Lösung der stationären inkompressiblen Navier-Stokes Gleichungen, Habilitationsschrift, Otto-von-Guericke-Universität, Magdeburg, 1997. (German) (1997) Zbl0915.76051
  16. 10.1051/m2an/1985190101111, RAIRO, Modélisation Math. Anal. Numér. 19 (1985), 111–143. (1985) MR0813691DOI10.1051/m2an/1985190101111
  17. Analysis of mixed finite element methods for the Stokes problem: a unified approach, Math. Comput. 42 (1984), 9–23. (1984) Zbl0535.76037MR0725982
  18. Efficient Solvers for Incompressible Flow Problems. An Algorithmic and Computational Approach, Springer-Verlag, Berlin, 1999. (1999) Zbl0930.76002MR1691839

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.